Unconventional correlated insulator in CrOCl-interfaced Bernal bilayer graphene

The realization of graphene gapped states with large on/off ratios over wide doping ranges remains challenging. Here, we investigate heterostructures based on Bernal-stacked bilayer graphene (BLG) atop few-layered CrOCl, exhibiting an over-1-GΩ-resistance insulating state in a widely accessible gate voltage range. The insulating state could be switched into a metallic state with an on/off ratio up to 107 by applying an in-plane electric field, heating, or gating. We tentatively associate the observed behavior to the formation of a surface state in CrOCl under vertical electric fields, promoting electron–electron (e–e) interactions in BLG via long-range Coulomb coupling. Consequently, at the charge neutrality point, a crossover from single particle insulating behavior to an unconventional correlated insulator is enabled, below an onset temperature. We demonstrate the application of the insulating state for the realization of a logic inverter operating at low temperatures. Our findings pave the way for future engineering of quantum electronic states based on interfacial charge coupling.

BLG has valley (+,-), spin( ,  ), plus the orbital (0,1, i.e., layer polarization), a total of 8-fold 'degeneracy' in the lowest Landau level. Experimentally, full degeneracy lifting has been observed in previous reports. [2,7] It is seen that all integer filling fraction of -1, -2, -3, -4,… can be found in those samples in the clean limit, which is similar to our results in Fig. 2 in the main text. The difference is, in the conventional BLG cases, energy levels in the zero energy Landau level (ZLL) are crossing in the space of D and n at a certain magnetic field B. This is not seen in our CrOCl-interfaced BLG samples, we attribute it to a high Deff range where crossing were not seen. While in the conventional case the interlayer potential u has to go through zero and crossings are inevitable at low energies. At this stage, we would tend to think that such valley symmetry broken phenomenon is unlikely related to the observed correlated insulator behavior. respectively. Data measured at T = 1.5 K, and the Deff was varied from -0.6 to 1.1 V/nm, with the ntot fixed at 2.2×10 12 cm -2 . Here, xx and xy are defined as xx=Rxx/(Rxx 2 +Rxy 2 ) and xy=Rxy/(Rxx 2 +Rxy 2 ), respectively. Four-probe lock-in measurements with a frequency of 17.77 Hz were used in this figure.

Supplementary Note 1. The electrostatic model of the h-BN/Bilayer Graphene (BLG)/CrOCl.
To understand the experimentally observed exotic features, we have in the following developed an electrostatic model in the studied system. We will first list two major unusual experimental observations as follows: 1. The 'iso-doping' line bending effect in the Deff-ntot mapping.
2. The large area of gapped state at charge neutrality of BLG, in the Deff-ntot mapping.
In the following, we will explain the above two major experimental observations using an electrostatic model. In short, the above observed 'insulating phase edge bending' and the 'huge area of charge neutral state' are attributed to the interfacial coupling between graphene and CrOCl surfaces. Such interfacial coupling invokes two mechanisms: 1) There exist interfacial states, which can serve as a reservoir of electrons with very large density of states (DOS), but have no free charges hence do not contribute to the transport directly.
2) A band structure reconstruction takes place once the Fermi level of graphene matches the lowest energy of the interfacial states, leading to a relative energy shift between interfacial states and BLG We assume that an interfacial charge layer exists in the h-BN/BLG/CrOCl heterostructure, which is located in bulk CrOCl below graphene with a distance of d2, as shown in Fig. 2c in the main text. The whole system can then be simplified into a capacitance model shown in Fig. 2d in the main text, with charge density defined as nt, n1 (i.e., nBLG), n2 (i.e., nCrOCl), nb, and the chemical potential defined as Vt(g), 0, Vb(g).
Here, t denotes top, b denotes bottom. Distance and dielectric constant of each dielectric layer are then written as d1, 1, d2, 1, d3, 3, with the inter-layer electric field written as E1, E2, E3. Moreover, the electron density in graphene and CrOCl layers are defined as 1, and 2, respectively.
Assuming that the interfacial states of BLG/CrOCl are located in the band gap of bulk CrOCl, and with high DOS and width, but do not contribute to transport. When BLG is tuned from the hole-side toward the CNP, i.e., Fermi level (EF) of BLG is lower than the energy of the lowest value of the interfacial states, only charges at the Fermi surface of graphene are at play in the transport. Hence everything is rather conventional, and the system behaves as 'pristine' BLG. The energy difference between CNP of BLG and the lowest band edge of the interfacial states is defined as E. It is worth mentioning that the band structure of BLG in our simplified model is parabolic instead of a "W" shape (or, often noted as the 'Mexican hat' shape) in reality. However, the modification of band structure has minor effect on the DOS and will not change the main result of our model.
When the BLG band is filled with charge carriers, its EF increases. At the point when EF matches the lowest value of the interfacial states, electrons start to fill into the DOS of interfacial states. We will argue in the coming parts that to observe the experimentally observed phenomena, there is an enlargement of E.
In the coming calculations, DOS for the interfacial states and BLG are written as, where we assume that the DOS of BLG is a constant Gr. The band gap gap is determined by external electric field E, the layer distance z0 of BLG, and the maximum band gap 1 as follows [10] : Here U = eEz0 (4) And E is the average electric field above and below the BLG, z0 ~ 0.33 nm is the layer distance of BLG. We noticed that, experimentally, d1/1 and d3/3 can be almost the same, and both d1 and d3 are way larger than d2. For simplicity, we define d2 = l (5) is a large dimensionless coefficient. Electron charge is written as -e, with e=1.6×10 -19 C.
Each electric field can be written as 1 1 Here, we omitted the potential between two layers of BLG since it is usually small compared with the applied voltage on top and bottom gates.
According to the Gauss's law ( ) and , 1, 2 ii n e i One obtains the following because the chemical potentials of the interfacial states and BLG are equal, one gets Here EF is the Fermi energy. Notice that in the above models, the difference of chemical potential of BLG caused by the movement of its Fermi level is omitted, as it is much smaller compared to the electrostatic energies induced by gate, and does not change the main results.
We then have two phases.: Phase-i (the 'conventional phase' defined in Fig. 2e-f in the main text), n2=0, and the heterostructure act like a "pristine" BLG, and

1=0.25e
Here we derive the DOS of BLG from its electron effective mass which is ~ 0.03 me, and a coefficient is multiplied to tune the model. In order to have the bending effect of CNP, we assume that E is an asymptotic function of n2, which may be a consequence of the bandwidth expansion due to electron interactions in the interfacial states. By taking a hypothesis of a simple asymptotic function, with A and B the fitting parameters: The iso-doping under the assumption of Eq. 17 can be plotted in Fig. 2f in the main text, with A ~ 6×10 20 and B ~ 1.5×10 -17 used in the calculations, respectively. It is worth mentioning that without the correction of Eq. 17, the calculated iso-doping lines in phase (ii) will be straight lines with no bending effect.
According to the above analysis, a diagram showing typical transition process in our system from Phase-i to Phase-ii is illustrated in Supplementary Figure 8. Data were obtained at zero magnetic field, and the temperature was elevated from 1.5 K to 100 K. Two-terminal Rxx was recorded using DC measurement. And each map was calibrated by subtracting the contact resistance Rc (measured by four-wire measurement at Vtg = 0 V and Vbg = -8 V for each temperature). It is clear that the extremely insulating phase at 1.5 K (orange color, with resistance > 1 G) shrinks its doping range of D, along with a decrease of the amplitude of resistance. The insulating phase prevails up to 100 K, but with shrunk phase area and lowered value of resistance.
Supplementary Figure 10. Dual gated map of Rxx of a typical BLG/CrOCl sample at T = 80 K. (a) DC sample resistance measured in a two-wire configuration in the parameter space of Vbg and Vtg at 80 K. It is seen that the insulating region is much shrunk as compared to that of, for example, T = 1.5 K. Also, the maximum resistance also reduces to a few tens of M, while it reached above 1 G at lower temperatures. Nevertheless, at this temperature (above liquid nitrogen temperature), the insulating phase is still way higher compared to those values reported elsewhere in dual gated bilayer graphene even at temperatures lower than 4 K. [6,[8][9] By extracting the maximums of the resistive peaks, we obtain the white circled line, as indicated in (a). Specifically, we chose 6 points (point-I to VI, indicated in (a)) in the insulating phase to perform thermal activation measurements, as shown in Supplementary It is clear that the critical voltage (where the system exits the insulating states and enters a resistive 'normal' state, with the relation of I-V becoming quasi-linear) is proportional to the distance L of electrodes, between which Vds is applied. Therefore, the critical voltage is actually denoting a critical in-plane electrical field E‖, defined as E‖ = Vds/L. c) Breakdown in-plane electric field Ec = Vc/L, plotted as a function of 1/L. The fact that the data points do NOT extrapolate to zero is a strong evidence that the current system is NOT of the Zener-type, but rather exhibits a pair-breaking behavior. [11] The dashed black line is guide to the eye. In other words, according to the relationship of Ec vs. 1/L in band insulators, the conventional Zener breakdown voltage Ec = Ec *L therefore should have little size-dependence, which is clearly ruled out in our system. It is also worth noting that the mesoscopic samples studied in our work exhibit insulating breakdown at Ec ~ 10 5 V/m, orders of magnitudes smaller than the values of Zener breakdown for band insulators (~ 10 8 V/m, which is size-independent). [12] This is in agreement with the theoretical model in in the limit of small L. [11] 9. Logic devices based on correlated insulator in BLG/CrOCl heterosystem. In this particular case, we used Device S22 and Device S26 in parallel, while a Vdd is biased through a resistor of 100 k. Notice that both devices are working in the NMOS-like regime, with Vtg for Device S22 and Device S26 set to be +7 V and +6 V, respectively. The inputs A and B for Device S22 and Device S26 are given as rectangular waveforms with the frequency of B doubled than that of A, and both amplitudes are 5 V. The waveforms of Vout (X value in (b)), input A, and input B are shown in (b) for different temperatures up to 80 K, which demonstrate a standard logic NAND function.